Disclaimer: obviously there will be spoilers about Lost Mine of Phandelver, as the question is tagged as such. I am not marking them as spoilers because the entire section becomes meaningless if you don't read the quoted parts.
There is some misunderstanding there. I will quote the actual book phrase:
In the unlikely event that the goblins defeat the adventurers, they leave them unconscious, loot them and the wagon, then head back to the Cragmaw hideout. The characters can continue to Phandalin, buy new gear at Barthen's Provisions, return to the ambush site, and find the goblin's trail.
In Barthen's Provisions, it states
Barthen's does not stock weapons or armor, but characters can purchase other adventuring gear here [...]
In the question, you mention
How do the Players get Gold to buy new Equipment?
Equipment usually refers to weapon and armor. The goblins DO NOT loot weapons and armor from the party. This is implied by the facts that the book says the party can get the new gear in Barthen's, which doesn't sell weapons and armor, and that gear is used to other adventurers' items from the PHB (torches and food, for example). That is, gear is not the same as equipment. If you want a justification for that: well, actually removing the armor from the party and then taking it away, while also carrying the content of the wagon, would be very hard for just 4 goblins. The equipment also might have no value for them, as they probably can't use it and provisions are better to steal/sell.
This should answer your question: they can still pretty much do everything, since they still have most of their combating gear. Eventually they will complete quests and get gold to be able to buy things back. I do agree that the "immediate" tone that the book describes is misleading, though, unless the goblins also don't loot the gold from the party.
Damn, I have already looted their weapons and armor! What do I do now??? - I noticed this might be the case after I had published my answer up to the first break.
Here are a few options:
Well, telling your players
So... it seems I screwed up and I was not supposed to take your weapon and armor away from you... It will make your life way harder than it should be... Are you fine with going back and pretending I did not take the equipment?
is not that bad.
Keep it going
If you really want to keep going from where you were, though, matsjoyce already gave you some ideas. You can make one (or more) NPCs more friendly to the party than they should be to strangers. The most justified one would be Barthen, IMHO, since he is friendly with Gundren (assuming you used the adventure hook from the book, he's the Dwarf that hired the party) and wants him rescued. Also, he is already a very friendly NPC. He could lend some money to the party (either pay in advance the 10gp/person or lend by trust) so they can rescue Gundren. LMoP gives lots of money so they should be able to pay him back soon.
I would not go with solutions that set precedents that you don't want to set. For example, from matsjoyce answer, scavaging dead people equipment could set the precedent for actually looting equipment from enemies, which you probably don't want to. The PHB (p. 144) mentions they are usually too damaged to sell. I would say they are also probably too damaged to use.
Weapons and armor used by monsters are rarely in good enough
condition to sell.